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Osamu Iyama

Publications and source records attributed to Osamu Iyama.

90 records · Page 5Linked to original sources
arXiv2010

n-representation-finite algebras and twisted fractionally Calabi-Yau algebras

In this short paper, we study $n$-representation-finite algebras from the viewpoint of the fractionally Calabi-Yau property. We shall show that all $n$-representation-finite algebras are twisted fractionally Calabi-Yau. We also show that for any $\ell>0$, twisted $\frac{n(\ell-1)}{\ell}$-Calabi-Yau algebras of global dimension at most $n$ are $n$-representation-finite. As an application, we give a construction of $n$-representation-finite algebras using the tensor product.

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arXiv2010

Cluster tilting for higher Auslander algebras

The concept of cluster tilting gives a higher analogue of classical Auslander correspondence between representation-finite algebras and Auslander algebras. The $n$-Auslander-Reiten translation functor $τ_n$ plays an important role in the study of $n$-cluster tilting subcategories. We study the category $\MM_n$ of preinjective-like modules obtained by applying $τ_n$ to injective modules repeatedly. We call a finite dimensional algebra $Λ$ \emph{$n$-complete} if $\MM_n=\add M$ for an $n$-cluster tilting object $M$. Our main result asserts that the endomorphism algebra $\End_Λ(M)$ is $(n+1)$-complete. This gives an inductive construction of $n$-complete algebras. For example, any representation-finite hereditary algebra $Λ^{(1)}$ is 1-complete. Hence the Auslander algebra $Λ^{(2)}$ of $Λ^{(1)}$ is 2-complete. Moreover, for any $n\ge1$, we have an $n$-complete algebra $Λ^{(n)}$ which has an $n$-cluster tilting object $M^{(n)}$ such that $Λ^{(n+1)}=\End_{Λ^{(n)}}(M^{(n)})$. We give the presentation of $Λ^{(n)}$ by a quiver with relations. We apply our results to construct $n$-cluster tilting subcategories of derived categories of $n$-complete algebras.

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arXiv2010

n-representation-finite algebras and n-APR tilting

We introduce the notion of n-representation-finiteness, generalizing representation-finite hereditary algebras. We establish the procedure of n-APR tilting, and show that it preserves n-representation-finiteness. We give some combinatorial description of this procedure, and use this to completely describe a class of n-representation-finite algebras called "type A".

math.RT↗
arXiv2010

Recollement of homotopy categories and Cohen-Macaulay modules

We study the homotopy category of unbounded complexes with bounded homologies and its quotient category by the homotopy category of bounded complexes. We show the existence of a recollement of the above quotient category and it has the homotopy category of acyclic complxes as a triangulated subcategory. In the case of the homotopy category of finitely generated projective modules over an Iwanaga-Gorenstein ring, we show that the above quotient category are triangle equivalent to the stable module category of Cohen-Macaulay $\opn{T}_2(R)$-modules.

math.RA↗
arXiv2009

Periodicities of T-systems and Y-systems

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of the Yangian or the quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for types A and C, and the direct method for types A, D, and B (level 2).

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arXiv2009

The Classification of Special Cohen-Macaulay Modules

In this paper we completely classify all the special Cohen-Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras.

math.AG↗
arXiv2008

Auslander-Reiten theory revisited

We recall several results in Auslander-Reiten theory for finite-dimensional algebras over fields and orders over complete local rings. Then we introduce $n$-cluster tilting subcategories and higher theory of almost split sequences and Auslander algebras there. Several examples are explained.

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arXiv2007

Mutation in triangulated categories and rigid Cohen-Macaulay modules

We introduce the notion of mutation of $n$-cluster tilting subcategories in a triangulated category with Auslander-Reiten-Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen-Macaulay modules over certain Veronese subrings.

math.RT↗
arXiv2007

Cluster structures for 2-Calabi-Yau categories and unipotent groups

We investigate cluster tilting objects (and subcategories) in triangulated 2-Calabi-Yau categories and related categories. In particular we construct a new class of such categories related to preprojective algebras of non Dynkin quivers associated with elements in the Coxeter group. This class of 2-Calabi-Yau categories contains the cluster categories and the stable categories of preprojective algebras of Dynkin graphs as special cases. For these 2-Calabi-Yau categories we construct cluster tilting objects associated with each reduced expression. The associated quiver is described in terms of the reduced expression. Motivated by the theory of cluster algebras, we formulate the notions of (weak) cluster structure and substructure, and give several illustrations of these concepts. We give applications to cluster algebras and subcluster algebras related to unipotent groups, both in the Dynkin and non Dynkin case.

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arXiv2007

Cluster tilting for one-dimensional hypersurface singularities

In this article we study Cohen-Macaulay modules over one-dimensional hypersurface singularities and the relationship with the representation theory of associative algebras using methods of cluster tilting theory. We give a criterion for existence of cluster tilting objects and their complete description by homological methods, using higher almost split sequences and results from birational geometry. We obtain a large class of 2-CY tilted algebras which are finite dimensional symmetric and satisfy $τ^2=\id$. In particular, we compute 2-CY tilted algebras for simple and minimally elliptic curve singularities.

math.RT↗
arXiv2007

Auslander-type conditions and cotorsion pairs

We study the properties of rings satisfying Auslander-type conditions. If an artin algebra $Λ$ satisfies the Auslander condition (that is, $Λ$ is an $\infty$-Gorenstein artin algebra), then we construct two kinds of subcategories which form functorially finite cotorsion pairs.

math.RA↗
arXiv2007

Fomin-Zelevinsky mutation and tilting modules over Calabi-Yau algebras

We say that an algebra $Λ$ over a commutative noetherian ring $R$ is Calabi-Yau of dimension $d$ ($d$-CY) if the shift functor $[d]$ gives a Serre functor on the bounded derived category of the finite length $Λ$-modules. We show that when $R$ is $d$-dimensional local Gorenstein the $d$-CY algebras are exactly the symmetric $R$-orders of global dimension $d$. We give a complete description of all tilting modules of projective dimension at most one for 2-CY algebras, and show that they are in bijection with elements of affine Weyl groups, preserving various natural partial orders. We show that there is a close connection between tilting theory for 3-CY algebras and the Fomin-Zelevinsky mutation of quivers (or matrices). We prove a conjecture of Van den Bergh on derived equivalence of non-commutative crepant resolutions.

math.RT↗
arXiv2006

Higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories

Auslander-Reiten theory is fundamental to study categories which appear in representation theory, for example, modules over artin algebras, Cohen-Macaulay modules over Cohen-Macaulay rings, lattices over orders, and coherent sheaves on projective curves. In these Auslander-Reiten theories, the number `2' is quite symbolic. For one thing, almost split sequences give minimal projective resolutions of simple functors of projective dimension `2'. For another, Cohen-Macaulay rings of Krull-dimension `2' provide us with one of the most beautiful situation in representation theory, which is closely related to McKay's observation on simple singularities. In this sense, usual Auslander-Reiten theory should be `2-dimensional' theory, and it be natural to find a setting for higher dimensional Auslander-Reiten theory from the viewpoint of representation theory and non-commutative algebraic geometry. We introduce maximal $(n-1)$-orthogonal subcategories as a natural domain of higher dimensional Auslander-Reiten theory which should be `$(n+1)$-dimensional'. We show that the $n$-Auslander-Reiten translation functor and the $n$-Auslander-Reiten duality can be defined quite naturally for such categories. Using them, we show that our categories have {\it $n$-almost split sequences}, which give minimal projective resolutions of simple objects of projective dimension `$n+1$' in functor categories. We show that an invariant subring (of Krull-dimension `$n+1$') corresponding to a finite subgroup $G$ of ${\rm GL}(n+1,k)$ has a natural maximal $(n-1)$-orthogonal subcategory. We give a classification of all maximal 1-orthogonal subcategories for representation-finite selfinjective algebras and representation-finite Gorenstein orders of classical type.

math.RT↗
arXiv2006

Auslander correspondence

We study Auslander correspondence from the viewpoint of higher dimensional Auslander-Reiten theory on maximal orthogonal subcategories. We give homological characterizations of Auslander algebras, especially an answer to a question of M. Artin. They are also closely related to Auslander's representation dimension of artin algebras and Van den Bergh's non-commutative crepant resolutions of Gorenstein singularities.

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arXiv2003

Rejective subcategories of artin algebras and orders

We will study the resolution dimension of functorially finite subcategories. The subcategories with the resolution dimension zero correspond to ring epimorphisms, and rejective subcategories correspond to surjective ring morphisms. We will study a chain of rejective subcategories to construct modules with endomorphisms rings of finite global dimension. We apply these result to study a function $r_Λ:\modΛ\to\nnn_{\ge0}$ which is a natural extension of Auslander's representation dimension.

math.RT↗
arXiv2003

Representation Dimension and Solomon Zeta Function

We will show that there exists a close relationship between quasi-hereditary algebras of Cline-Parshall-Scott and the rejection theory from the viewpoint of the approximation theory of Auslander-Smalo. As an application, we will solve two open problems. One concerns the representation dimension of artin algebras introduced by M. Auslander about 30 years ago, and another concerns the Solomon zeta functions of orders introduced by L. Solomon about 25 years ago. It will turn out that the rejection theory relates these two quite different problems with each other closely.

math.RA↗
arXiv2002

Radical embeddings and representation dimension

Given a representation-finite algebra B and a subalgebra A of B such that the Jacobson radicals of A and B coincide, we prove that the representation dimension of A is at most three. By a result of Igusa and Todorov, this implies that the finitistic dimension of A is finite.

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